MR. IVORY ON THE THEORY OF THE ELLIPTIC TRANSCENDENTS. 353 
If we admit that this is a possible equation, and suppose that when ^ is succes- 
sively equal to the arcs of the series, 
0, "2“ , - “2" ’ ^ , &c., 
attains the respective values, 
0, Xj, X 2 , X 3 , &C. ; 
we shall have, 
H = (3 K (Xj), 2 H = (3 K (X. 2 ), 3 H = (3 K (X 3 ), &c. ; 
and consequently, 
K(X 2 )= 2 K(\ 1 ), KW = 3 KW,&c. 
Thus the arcs X 2 , X 3 , &c. are the amplitudes of the multiples of the function 
K (>4), which itself remains indeterminate. We may therefore suppose 
p X K (X x ) = K (y)> P representing any integer number; and, in conse- 
quence, we shall have 
K(X.) = j K, K(A 2 ) = J-K, ... K(X„) = ~K. 
Any proposed number being assumed for p, we may determine the amplitudes 
X 1} X 2 , ?. 3 , &c. by the theory for the multiplication and subdivision of elliptic 
functions : but as the equations to be solved are complicated and impracticable, 
the arcs X l5 X 2 , &c. may be treated as known quantities without any attempt to 
compute them. 
An elliptic function becomes equal to the arc of its amplitude, when the 
modulus vanishes : and in this case the arcs X l5 X 2 , X 3 , &c. are obtained by the 
1 7T 
subdivision of the quadrant of the circle, and are respectively equal to — . — , 
2 7T 3 TT o 
~p * ~p • &C ' 
Having made these observations, we shall for the present dismiss all consi- 
deration of the equation to be demonstrated, and turn our attention to inves- 
tigate two variable arcs ^ and <p, such that the first shall have the successive 
values, 
/-v 7T « 
2-,,2X 3 x &c. 
2 z 2 
